Multinomial Coefficient

combinatorics

Definition

Multinomial Coefficient

Let be non-negative integers satisfying

The multinomial coefficient

counts the ways to assign distinct positions to labelled groups so that group receives exactly positions. Equivalently, it counts permutations with repetition of a multiset whose distinct values have multiplicities .

Sequential Factorisation

Choose the positions of the first group, then the positions of the second group from those remaining, and continue until the last group is forced. The product rule gives the canonical factorisation

Expanding the binomial coefficients makes the intermediate factorials telescope, leaving

Example

Two copies of and one copy of

Choose two of the three positions for ; the remaining position is forced to contain .

Therefore

Binomial Special Case

For and , the multinomial coefficient reduces to a binomial coefficient:

Multinomial Expansion

The coefficients in the expansion of are multinomial coefficients: